1. Field of the Invention
The present invention relates to an electromagnetic field simulator for simulating an electromagnetic field, an electromagnetic field analysis system for analyzing an electromagnetic field, an electromagnetic field simulating program product for using a computer to simulate an electromagnetic field, and an electromagnetic field simulating program product for using a computer to analyze an electromagnetic field, and more specifically to a technique of correctly modeling an electromagnetic wave irradiated to an analysis area.
2. Description of the Related Art
Conventionally, to analyze the performance and the like of a communication antenna, an electromagnetic field simulator and an electromagnetic field simulating program for calculating an electromagnetic field for a communication device and the like are well known. As a typical method which is applied the electromagnetic field simulator and the electromagnetic field simulating program for analyzing an electromagnetic field, a finite difference time domain method (FDTD method) is well known (for example, the documents 1 and 2 described below).
The method is a finite difference calculation of Maxwell's equations which is basic equations of the electromagnetic field, and is used for the calculation of the electromagnetic field. In this method, the change of the electromagnetic field is simulated in detail by setting a sufficiently short time step and a small space between grids. The merits of the FDTD method are a high-speed performance because of a simple calculation, a transitional analysis of the electromagnetic wave by a calculation of a waveform, and suitable for 3-dimensional calculation, etc.
The Maxwell's equations are configured by the following equations 1 and 2.
                                          ∇                          ×              H                                -                      ɛ            ⁢                                          ∂                E                                            ∂                t                                                    =                  σ          ⁢                                          ⁢          E                                    (                  equation          ⁢                                          ⁢          1                )                                                      ∇                          ×              E                                +                      μ            ⁢                                          ∂                H                                            ∂                t                                                    =        0                            (                  equation          ⁢                                          ⁢          2                )                            where E indicates an electric field vector, H indicates an electromagnetic field vector, ε indicates a permittivity, σ indicates a conductivity, and μ indicates a permeability.        
The equations 3 and 4 which are obtained from the equation 1 and 2 to calculate the change of an electromagnetic field component are shown below. (refer to Allen Taflove, Suan C. Hagness: “Computational Electrodynamics” (ARTECH HOUCE, INC.)).
                                          E            x                    ⁢                                    |                                                      i                +                                  1                  /                  2                                            ,              j              ,              k                        n                          =                                            (                                                1                  -                                                                                    σ                                                                              i                            +                                                          1                              /                              2                                                                                ,                          j                          ,                          k                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                                              2                      ⁢                                              ɛ                                                                              i                            +                                                          1                              /                              2                                                                                ,                          j                          ,                          k                                                                                                                                      1                  +                                                                                    σ                                                                              i                            +                                                          1                              /                              2                                                                                ,                          j                          ,                          k                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                                              2                      ⁢                                              ɛ                                                                              i                            +                                                          1                              /                              2                                                                                ,                          j                          ,                          k                                                                                                                                )                        ⁢                          E              x                        ⁢                                          ❘                                                              i                  +                                      1                    /                    2                                                  ,                j                ,                k                                            n                -                1                                              +                                    (                                                                    Δ                    ⁢                                                                                  ⁢                    t                                                        ɛ                                                                  i                        +                                                  1                          /                          2                                                                    ,                      j                      ,                      k                                                                                        1                  +                                                                                    σ                                                                              i                            +                                                          1                              /                              2                                                                                ,                          j                          ,                          k                                                                    ⁢                      Δ                      ⁢                                                                                          ⁢                      t                                                              2                      ⁢                                              ɛ                                                                              i                            +                                                          1                              /                              2                                                                                ,                          j                          ,                          k                                                                                                                                )                        ·                          (                                                                                          H                      z                                        ⁢                                          ❘                                                                        i                          +                                                      1                            /                            2                                                                          ,                                                  j                          +                                                      1                            /                            2                                                                          ,                        k                                                                    n                        -                                                  1                          /                          2                                                                                      ⁢                                                                  -                                                  H                          z                                                                    ⁢                                              ❘                                                                              i                            +                                                          1                              /                              2                                                                                ,                                                      j                            -                                                          1                              /                              2                                                                                ,                          k                                                                          n                          -                                                      1                            /                            2                                                                                                                                                    Δ                    ⁢                                                                                  ⁢                    y                                                  -                                                                            H                      y                                        ⁢                                          ❘                                                                        i                          +                                                      1                            /                            2                                                                          ,                        j                        ,                                                  k                          +                                                      1                            /                            2                                                                                                                      n                        -                                                  1                          /                          2                                                                                      ⁢                                                                  -                                                  H                          y                                                                    ⁢                                                                        ❘                                                                                                      i                            +                                                          1                              /                              2                                                                                ,                          j                          ,                                                      k                            -                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                                                    Δ                    ⁢                                                                                  ⁢                    z                                                              )                                                          (                  equation          ⁢                                          ⁢          3                )                                                      H            x                    ⁢                      ❘                          i              ,                              j                +                                  1                  /                  2                                            ,                              k                +                                  1                  /                  2                                                                    n              +                              1                /                2                                                    =                              H            x                    ⁢                      ❘                          i              ,                              j                +                                  1                  /                  2                                            ,                              k                +                                  1                  /                  2                                                                    n              -                              1                /                2                                              ⁢                                    +                              (                                                      Δ                    ⁢                                                                                  ⁢                    t                                                        μ                                          i                      ,                                              j                        +                                                  1                          /                          2                                                                    ,                                              k                        +                                                  1                          /                          2                                                                                                                    )                                      ·                          (                                                                                          E                      y                                        ⁢                                          ❘                                              i                        ,                                                  j                          +                                                      1                            /                            2                                                                          ,                                                  k                          +                          1                                                                    n                                        ⁢                                                                  -                                                  E                          y                                                                    ⁢                                              ❘                                                  i                          ,                                                      j                            +                                                          1                              /                              2                                                                                ,                          k                                                n                                                                                                  Δ                    ⁢                                                                                  ⁢                    z                                                  -                                                                            E                      z                                        ⁢                                          ❘                                              i                        ,                                                  j                          +                          1                                                ,                                                  k                          +                                                      1                            /                            2                                                                                              n                                        ⁢                                                                  -                                                  E                          z                                                                    ⁢                                              ❘                                                  i                          ,                          j                          ,                                                      k                            +                                                          1                              /                              2                                                                                                      n                                                                                                  Δ                    ⁢                                                                                  ⁢                    y                                                              )                                                          (                  equation          ⁢                                          ⁢          4                )                            where Ex, Ey, and Ez indicate each component in an electric field, Hx, Hy, and Hz indicate each component in an electromagnetic field, ε indicates a permittivity, σ indicates a conductivity, Δx, Δy, and Δz indicate a space grid width, and Δt indicates the width of a time division.        
The subscripts i, j, and k at the lower right of the electromagnetic field component are the numbers of a space grid. They indicate the coordinates of a simulation space.
FIG. 1 shows the arrangement of each component of an electric field and a magnetic field with the FDTD method.
FIG. 1 shows one cell obtained by dividing an analysis model to which the FDTD method is applied, and the space arrangement of the component of each electric field and magnetic field in one cell is shown. The subscripts n and n+1 at the upper right of the electromagnetic field component are the time steps. When the time step advances 1, the time advances Δt. By sequentially repeating the equations 3 and 4, the change of an electromagnetic field in the 3-dimensional space can be calculated.
Since the FDTD method performs a calculation with the finite difference method, it is necessary to set the space between the grids on short distance about 1/10˜ 1/20 of the wavelength of the electromagnetic wave. During the calculation, the momentary values of the electromagnetic fields in a whole of the analysis model are stored in the main storage such as RAM (random access memory) and the like. Since the storage capacity of a computer is restricted, the size of a model must be generally within several times larger than the wavelength. Since the wavelength of light is equal to or less than 1 micrometer, the analysis area with the FDTD method is within several micrometers. For that reason there have been few cases in which the FDTD method is applied to the optical simulation.
Recently, near-field optics using an optical component having a structure smaller than the wavelength of light has attracted an attention. Because the diffraction has a serious influence on the light wave propagation in the near-field optics, it is impossible to use a ray tracing technique. In this case, the light is processed as an electromagnetic wave, and the propagation calculation of the light is performed in the same method as in the analysis of an electric wave. The analysis is originally a 3-dimensional calculation, and an analysis area is several times larger than that of the wavelength. Therefore, the FDTD method is effective. The study of the near-field optics and the development of an optical device with the FDTD method have been actively performed.
When an optical analysis is performed with the FDTD method, the analysis is frequently performed with the light irradiated from a outside of an analysis area because the analysis area is within several micrometers for the above-mentioned reason. The analysis including the light source such as a laser device and the like cannot be performed. For incident light is reproduced, the electromagnetic field of a part of an analysis area are vibrated by the frequency of the light. If the vibration of the electric field executed with the repetition of the calculation of the equations 3 and 4, the vibration propagates to the surrounding electromagnetic fields, thereby propagating the electromagnetic wave, that is, simulating the propagation of the light. The area in which an electromagnetic field is excited is referred to as a wave source because an electromagnetic wave is generated from the area. The wave source is virtually set to execute a calculation, but does not practically exist in an actual analysis object.
To perform a correct analysis, it is important to correctly model the light propagating from the wave source. For example, if the wavelength, the direction, and the polarization state of incident light satisfy a predetermined condition with respect to a small particle, thin film, etc. in nanometer size, light is located locally with Plasmon resonance, which is a well known phenomenon. Furthermore, since common optical phenomena such as reflection, refraction, etc. also largely depend on the conditions of incident light, correct incident light settings are required.
Especially required in setting the incident light is the function of impinging a plane wave with a uniform intensity distribution at an arbitrary angle into an object. A plane wave refers to the phase of an electromagnetic wave aligned on the perpendicular plane to the travel direction.
In a practical experiment system, a beam launched from a laser device is normally some millimeters in diameter, and when the beam is focused by a single lens, it is still several tens of micrometers in diameter. Therefore, in a narrow analysis area of several micrometers to be applied the FDTD method, uniform intensity distribution of the incident beam is an accurate analysis condition for realizing a simulation. In addition, it is necessary to confirm the accuracy of calculations. However, considering the comparisons with other calculating methods, a plane wave with uniform intensity distribution is assumed inmost cases in an analytic calculation performed by solving an equation. Therefore, it is also necessary to perform a calculation for comparison by setting a plane wave with uniform intensity distribution in the FDTD method.
However, incident light cannot be set as a plane wave with uniform intensity distribution only by simply exciting an electric field and a magnetic field in a perpendicular plane area to the electromagnetic wave propagation direction.
FIG. 2 shows an electric field distribution of an electromagnetic wave from a wave source on which an electric field is excited with a uniform amplitude and phase on one excitation plane.
Since an electromagnetic wave is generated on both side of a wave source on which only the electric field is excited, the electromagnetic wave is made to propagate only in one direction by exciting the magnetic field as well as the electric field in the calculation shown in FIG. 2. While the excitation condition is based on modeling a plane wave, a propagating electromagnetic wave is non-uniform intensity distribution, and spreads in the ±X directions, thereby failing in generating a plane wave because a diffraction phenomenon occurs in the electromagnetic wave.
Since the FDTD method is directly calculation in the entire analysis space, interference and diffraction are reproduced as in natural phenomena. In FIG. 2, since the electric field amplitude is kept constant on the excitation plane, intense diffraction is caused by the same phenomenon as a cutoff electromagnetic wave at the edge of the excitation plane. To realize a plane wave with uniform intensity distribution in the FDTD method, it is necessary to take measures against the diffraction.
The simplest method for suppressing the diffraction is to set the intensity distribution on the Gaussian distribution, and set the electric field intensity on the excitation plane boundary to 0. If the distribution width is sufficiently large, it is assumed that uniform intensity distribution has been approximately obtained. Since this method can be easily realized by changing the intensity distribution without changing the configuration of the excitation plane. However, there are the problems with this method that the uniform intensity distribution cannot be realized only approximately, and a large calculation area is required.
Although diffraction does not occur at the edge of the excitation area in the Gaussian distribution, the influence of diffraction develops when the distribution width is as small as the wavelength, and the electromagnetic wave changes into a spreading wave. Although the diffraction can be smaller if the distribution width is large, an excitation plane exceeding the wavelength is set even if a target object is smaller than the wavelength of incident light and is some nanometers in size, thereby largely increasing the calculation time and required memory size. To efficiently perform an analysis, a wave source that suppresses diffraction for any small excitation area is required.
To realize incident light of uniform intensity by suppressing a diffraction phenomenon, there is a method described in the following document 3.
FIG. 3 shows the electric field intensity distribution of an electromagnetic wave propagating from the wave source configured by six excitation planes.
As shown in FIG. 3, diffraction can be suppressed by enclosing the area in which the electromagnetic wave propagates, and exciting the electric fields and the electromagnetic fields of all excitation planes on the basis of the propagating electromagnetic wave.
Practically described below is the method of suppressing diffraction.
FIG. 4 shows the conventional wave source configured by six excitation planes.
As shown in FIG. 4, a wave source configured by six excitation planes for an analysis space is assumed to consider a case in which an electromagnetic wave propagates in the Z direction from an excitation plane 1 perpendicular to the travel direction of the electromagnetic wave.
Assume that the electromagnetic field of an electromagnetic wave vibrates only in the X direction. Excitation planes 2 through 5 perpendicularly contacting the four sides of the excitation plane 1 are arranged. The excitation planes 2 and 3 are perpendicular to the X-axis, and the excitation planes 4 and 5 are perpendicular to the Y-axis. To suppress the diffraction, the electric fields and the electromagnetic fields of the excitation planes 2 through 5 are excited so that an electromagnetic wave can propagate inside the area enclosed by the excitation planes 1 through 6, the electromagnetic field can be 0 outside the area, and it cannot affect the inside.
First consider the excitation plane 2 perpendicular to the X-axis.
The number of the grid in the X direction on the excitation plane 2 is assigned i1, and it is assumed that an electromagnetic wave propagates in an area having a larger grid number. The components arranged on the excitation planes 2 and 3 are Ey, Ez, and Hx. The equations for calculating the three components in the FDTD method are expressed as the following equations 5, 6, and 7.
                              Ey          ⁢                                    ❘                                                      i                1                            ,                              j                +                                  1                  /                  2                                            ,              k                        n                          =                                                            Cy                                                      i                    1                                    ,                                      j                    +                                          1                      /                      2                                                        ,                  k                                            ·              Ey                        ⁢                                          ❘                                                              i                  1                                ,                                  j                  +                                      1                    /                    2                                                  ,                k                                            n                -                1                                              +                                    Gy                                                i                  1                                ,                                  j                  +                                      1                    /                    2                                                  ,                k                                      ·                          {                                                                                          Hx                      ⁢                                                                        ❘                                                                                                      i                            1                                                    ,                                                      j                            +                                                          1                              /                              2                                                                                ,                                                      k                            +                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                -                                          Hx                      ⁢                                                                        ❘                                                                                                      i                            1                                                    ,                                                      j                            +                                                          1                              /                              2                                                                                ,                                                      k                            -                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                                                    Δ                    ⁢                                                                                  ⁢                    z                                                  -                                                                            Hz                      ⁢                                                                        ❘                                                                                                                                    i                              1                                                        +                                                          1                              /                              2                                                                                ,                                                      j                            +                                                          1                              /                              2                                                                                ,                          k                                                                          n                          -                                                      1                            /                            2                                                                                                                -                                          Hz                      ⁢                                                                        ❘                                                                                                                                    i                              1                                                        -                                                          1                              /                              2                                                                                ,                                                      j                            +                                                          1                              /                              2                                                                                ,                          k                                                                          n                          -                                                      1                            /                            2                                                                                                                                                    Δ                    ⁢                                                                                  ⁢                    x                                                              }                                                          (                  equation          ⁢                                          ⁢          5                )                                          Ez          ⁢                                    ❘                                                      i                1                            ,              j              ,                              k                +                                  1                  /                  2                                                      n                          =                                            Cz                                                i                  1                                ,                j                ,                                  k                  +                                      1                    /                    2                                                                        ⁢            Ez            ⁢                                          ❘                                                              i                  1                                ,                j                ,                                  k                  +                                      1                    /                    2                                                                              n                -                1                                              +                                    Gz                                                i                  1                                ,                j                ,                                  k                  +                                      1                    /                    2                                                                        ·                          {                                                                                          Hy                      ⁢                                                                        ❘                                                                                                                                    i                              1                                                        +                                                          1                              /                              2                                                                                ,                          j                          ,                                                      k                            +                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                -                                          Hy                      ⁢                                                                        ❘                                                                                                                                    i                              1                                                        -                                                          1                              /                              2                                                                                ,                          j                          ,                                                      k                            +                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                                                    Δ                    ⁢                                                                                  ⁢                    x                                                  -                                                                            Hx                      ⁢                                                                        ❘                                                                                                      i                            1                                                    ,                                                      j                            +                                                          1                              /                              2                                                                                ,                                                      k                            +                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                -                                          Hx                      ⁢                                                                        ❘                                                                                                      i                            1                                                    ,                                                      j                            -                                                          1                              /                              2                                                                                ,                                                      k                            +                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                                                    Δ                    ⁢                                                                                  ⁢                    y                                                              }                                                          (                  equation          ⁢                                          ⁢          6                )                                          Hx          ⁢                                    ❘                                                      i                1                            ,                              j                +                                  1                  /                  2                                            ,                              k                +                                  1                  /                  2                                                                    n              +                              1                /                2                                                    =                              Hx            ⁢                                          ❘                                                              i                  1                                ,                                  j                  +                                      1                    /                    2                                                  ,                                  k                  +                                      1                    /                    2                                                                              n                -                                  1                  /                  2                                                              +                                    Bx                                                i                  ⁢                                                                          ⁢                  1                                ,                                  j                  +                                      1                    /                    2                                                  ,                                  k                  +                                      1                    /                    2                                                                        (                                                                                Ey                    ⁢                                                                  ❘                                                                                              i                          1                                                ,                                                  j                          +                                                      1                            /                            2                                                                          ,                                                  k                          +                          1                                                                    n                                                        -                                      Ey                    ⁢                                                                  |                                                                                              i                          1                                                ,                                                  j                          +                                                      1                            /                            2                                                                          ,                        k                                            n                                                                                        Δ                  ⁢                                                                          ⁢                  z                                            -                                                                    Ez                    ⁢                                                                  ❘                                                                                              i                          1                                                ,                                                  j                          +                          1                                                ,                                                  k                          +                                                      1                            /                            2                                                                                              n                                                        -                                      Ez                    ⁢                                                                  ❘                                                                                              i                          1                                                ,                        j                        ,                                                  k                          +                                                      1                            /                            2                                                                                              n                                                                                        Δ                  ⁢                                                                          ⁢                  y                                                      )                                              (                  equation          ⁢                                          ⁢          7                )            
To simplify the expression, the factors are collectively defined as Cy, Gy, Cz, Gz, and Bx. The equation 7 is the same as the equation 4.
If By, Ez, and Hx can be constantly 0 with respect to the travel of an electromagnetic wave, the area in which the electromagnetic wave does not propagates is constantly 0, thereby causing no diffraction. Since the propagating electromagnetic wave is X-polarized in the Z direction, Ey, Ez, Hx, and Hz are 0, and Ex and Hy have values other than 0.
An equation including Hy is the equation 6. This equation includes two Hy, that is, Hy (i1+½, j, k+½) and Hy (i1−½, j, k+½). Since the grid number i1+½ in the X direction belongs to the area in which the electromagnetic wave propagates, it has a value other than 0. Since i1−½ is outside the area, it has the value of 0. Therefore, the value of 0 can be obtained by calculating Hy when the electromagnetic wave propagates and subtracting the result from Ez. That is, the following equation 8 is executed.
                              Ez          ⁢                                    ❘                                                      i                1                            ,              j              ,                              k                +                                  1                  /                  2                                                      n                          =                              Ez            ⁢                                          ❘                                                              i                  1                                ,                j                ,                                  k                  +                                      1                    /                    2                                                              n                                -                                    G                                                i                  1                                ,                j                ,                                  k                  +                                      1                    /                    2                                                                        ⁢                                                            Hy                  wave                                ⁢                                                      ❘                                                                                                      i                        1                                            +                                              1                        /                        2                                                              ,                    j                    ,                                          k                      +                                              1                        /                        2                                                                                                  n                    +                                          1                      /                      2                                                                                                  Δ                ⁢                                                                  ⁢                x                                                                        (                  equation          ⁢                                          ⁢          8                )            
The Hy wave is an electromagnetic field component of the electromagnetic wave, and is calculated by the following equation 9.Hywave|i1+1/2,j,k+1/2n−1/2=Z·E0 sin [ω(n− 1/2)Δt+φi1+1/2,j,k+1/2]  (equation 9)                where E0 indicates the amplitude of the electric field of the electromagnetic wave, Z indicates the impedance of the space, φi1+1/2,j,k+1/2 indicates the phase of the electromagnetic field in the grid coordinates (i1+½,j,k+½), and ω indicates the angular frequency of the electromagnetic wave.        
When Ez is forcibly set to 0 without performing the procedure, the electromagnetic field calculation is not performed on the plane. Therefore, the electromagnetic wave is reflected. To avoid it, the equation 8 for erasing the electromagnetic field Hy is to be performed after first applying the equation 6. The same concept can be applied to the excitation plane 3. Since the inside and the outside are reversed, the arithmetic of adding the Hy wave calculated from the propagation of the electromagnetic wave is performed.
Next, the excitation plane 4 perpendicular to the Y-axis is considered.
Since the excitation electric field Ex of the electromagnetic wave is located at the grid number j1 in the Y direction, the coordinates set to 0 are located j1−½ at a half cell outside. The components of j1−½ plane are Ey, Hx, and Hz, and the respective equations are the following equations 10, 11, and 12.
                              Ey          ⁢                                    ❘                                      i              ,                                                j                  1                                -                                  1                  /                  2                                            ,              k                        n                          =                                                            Cy                                  i                  ,                                                            j                      1                                        -                                          1                      /                      2                                                        ,                  k                                            ·              Ey                        ⁢                                          ❘                                            i                ,                                                      j                    1                                    -                                      1                    /                    2                                                  ,                k                                            n                -                1                                              +                                    Gy                              i                ,                                                      j                    1                                    -                                      1                    /                    2                                                  ,                k                                      ·                          {                                                                                          Hx                      ⁢                                                                        ❘                                                                          i                          ,                                                                                    j                              1                                                        -                                                          1                              /                              2                                                                                ,                                                      k                            +                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                -                                          Hx                      ⁢                                                                        ❘                                                                          i                          ,                                                                                    j                              1                                                        -                                                          1                              /                              2                                                                                ,                                                      k                            -                                                          1                              /                              2                                                                                                                                n                          -                                                      1                            /                            2                                                                                                                                                    Δ                    ⁢                                                                                  ⁢                    z                                                  -                                                                            Hz                      ⁢                                                                        ❘                                                                                                      i                            +                                                          1                              /                              2                                                                                ,                                                                                    j                              1                                                        -                                                          1                              /                              2                                                                                ,                          k                                                                          n                          -                                                      1                            /                            2                                                                                                                -                                          Hz                      ⁢                                                                        ❘                                                                                                      i                            -                                                          1                              /                              2                                                                                ,                                                                                    j                              1                                                        -                                                          1                              /                              2                                                                                ,                          k                                                                          n                          -                                                      1                            /                            2                                                                                                                                                    Δ                    ⁢                                                                                  ⁢                    x                                                              }                                                          (                  equation          ⁢                                          ⁢          10                )                                          Hx          ⁢                                    ❘                                      i              ,                                                j                  1                                -                                  1                  /                  2                                            ,                              k                +                                  1                  /                  2                                                                    n              +                              1                /                2                                                    =                              Hx            ⁢                                          ❘                                            i                ,                                                      j                    1                                    -                                      1                    /                    2                                                  ,                                  k                  +                                      1                    /                    2                                                                              n                -                                  1                  /                  2                                                              +                                    Bx                              i                ,                                                      j                    1                                    -                                      1                    /                    2                                                  ,                                  k                  +                                      1                    /                    2                                                                        (                                                                                Ey                    ⁢                                                                  ❘                                                                    i                        ,                                                                              j                            1                                                    -                                                      1                            /                            2                                                                          ,                                                  k                          +                          1                                                                    n                                                        -                                      Ey                    ⁢                                                                  ❘                                                                    i                        ,                                                                              j                            1                                                    -                                                      1                            /                            2                                                                          ,                        k                                            n                                                                                        Δ                  ⁢                                                                          ⁢                  z                                            -                                                                    Ez                    ⁢                                                                  ❘                                                                    i                        ,                                                  j                          1                                                ,                                                  k                          +                                                      1                            /                            2                                                                                              n                                                        -                                      Ez                    ⁢                                                                  ❘                                                                    i                        ,                                                                              j                            1                                                    -                          1                                                ,                                                  k                          +                                                      1                            /                            2                                                                                              n                                                                                        Δ                  ⁢                                                                          ⁢                  y                                                      )                                              (                  equation          ⁢                                          ⁢          11                )                                          Hz          ⁢                                    ❘                                                      i                +                                  1                  /                  2                                            ,                                                j                  1                                -                                  1                  /                  2                                            ,              k                                      n              +                              1                /                2                                                    =                              Hz            ⁢                                          ❘                                                              i                  +                                      1                    /                    2                                                  ,                                                      j                    1                                    -                                      1                    /                    2                                                  ,                k                                            n                -                                  1                  /                  2                                                              +                                    Bz                                                i                  +                                      1                    /                    2                                                  ,                                                      j                    1                                    -                                      1                    /                    2                                                  ,                k                                      (                                                                                Ex                    ⁢                                                                  ❘                                                                                              i                          +                                                      1                            /                            2                                                                          ,                                                  j                          1                                                ,                        k                                            n                                                        -                                      Ex                    ⁢                                                                  ❘                                                                                              i                          +                                                      1                            /                            2                                                                          ,                                                                              j                            1                                                    -                          1                                                ,                        k                                            n                                                                                        Δ                  ⁢                                                                          ⁢                  y                                            -                                                                    Ey                    ⁢                                                                  ❘                                                                                              i                          +                          1                                                ,                                                                              j                            1                                                    -                                                      1                            /                            2                                                                          ,                        k                                            n                                                        -                                      Ey                    ⁢                                                                  ❘                                                                    i                        ,                                                                              j                            1                                                    -                                                      1                            /                            2                                                                          ,                        k                                            n                                                                                        Δ                  ⁢                                                                          ⁢                  x                                                      )                                              (                  equation          ⁢                                          ⁢          12                )            
In the equations above, only the component Ex has a value other than 0, and the corresponding equation is the equation 12. Therefore, if Ex is calculated from the propagation of the electromagnetic wave and the related terms are removed from the equation 12, then Hz can be set to 0.
                                                                        Hz                ⁢                                  ❘                                                            i                      +                                              1                        /                        2                                                              ,                                                                  j                        1                                            -                                              1                        /                        2                                                              ,                    k                                                        n                    +                                          1                      /                      2                                                                                  =                            ⁢                              Hz                ⁢                                  ❘                                                            i                      +                                              1                        /                        2                                                              ,                                                                  j                        1                                            -                                              1                        /                        2                                                              ,                    k                                                        n                    +                                          1                      /                      2                                                                      +                                                                                                      ⁢                                                Bz                                                            i                      +                                              1                        /                        2                                                              ,                                                                  j                        1                                            -                                              1                        /                        2                                                              ,                    k                                                  ⁢                                                                            Ex                      wave                                        ⁢                                          ❘                                                                        i                          +                                                      1                            /                            2                                                                          ,                                                                              j                            1                                                    -                          1                                                ,                        k                                            n                                                                            Δ                    ⁢                                                                                  ⁢                    y                                                                                                          (                  equation          ⁢                                          ⁢          13                )                                                      Ex            wave                    ⁢                      ❘                                          i                +                                  1                  /                  2                                            ,                                                j                  1                                -                1                            ,              k                        n                          =                              E            0                    ⁢                      sin            ⁡                          [                                                ω                  ⁢                                                                          ⁢                  n                  ⁢                                                                          ⁢                  Δ                  ⁢                                                                          ⁢                  t                                +                                  ϕ                                                            i                      +                                              1                        /                        2                                                              ,                                                                  j                        1                                            -                      1                                        ,                    k                                                              ]                                                          (                  equation          ⁢                                          ⁢          14                )                            where φi+½,j1−1,k indicates the phase of the electric field in the grid coordinates (i+½, j−1,k).        
Described above is the case in which an electromagnetic wave travels parallel to the coordinates axis. It is also possible to generate a plane wave of uniform intensity by avoiding the diffraction under the similar concept when diagonal travel is performed.
As described above, in the method of enclosing by the excitation planes the area in which the electromagnetic wave propagates, the calculation performed when a plane wave enters the microscatterer shown in FIG. 5.
FIG. 5 is an explanatory view of the calculation of a scattering electromagnetic wave by a microsphere using the conventional wave source.
FIG. 5 shows a calculation value of the electric field intensity distribution when the light having the wavelength of 400 nanometers enters a gold of 200 nanometers in diameter using the above-mentioned conventional wave source. The incident wave is a plane wave of correctly uniform intensity. The analysis area can be a little larger than the excitation plane enclosing the microscatterer, and the calculation can be performed in a small space. The example of the calculation shown in FIG. 5 shows that the conventional wave source can be effectively applied about the scattering of water vapor and fine particles in the atmosphere.    Document 1: Yee, K. S., “Numerical solution of initial boundary value problems involving Maxwell's equations” in isotropic media”, IEEE Trans. Antennas and Propagation., Vol. Ap-14, p. 302-307, 1966.    Document 2: Toru Uno, “Electromagnetic Field and Antenna Analysis in FDTD Method”, 1998, Corona    Document 3: Allen Taflove, Susan C. Hagness: “Computational Electrodynamics” (ARTECH HOUSE, INC.)
However, in the near-field optics, a scatterer is arranged on a substrate such as glass, silicon, etc. to launch light into the substrate. FIG. 6 shows the result of calculating the analysis model using the wave source explained above in the FDTD method.
FIG. 6 shows the electric field intensity distribution obtained by launching the electromagnetic wave at the angle of incident of 45° into the reflection plane as a result of calculation using the conventional wave source.
As shown in FIG. 6, the normal of the silicon substrate is arranged to overlap the Z-axis, the electromagnetic wave of the wavelength of 400 nanometers enters at the angle of incident of 45° into the YZ plane as an incident plane. The plane contacting the substrate is not excited, but fine planes enclosing the reflection plane are excited. In FIG. 6, the electric field intensity distribution is expressed as a three-view drawing by the portion (a) as the YZ plane, the portion (b) as the ZX plane, and the portion (c) as the XY plane. The intensity distribution is non-uniform, and there is the problem that the uniformity in intensity, which is the purpose of configuring a wave source by a plurality of excitation planes, cannot be realized.